Subgroups — #36537858

Order 22784.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031431	C₂	1	2	yes	maximal
2	#26029227	C₂	1	1	yes	maximal, characteristic
3	#26019777	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019772	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019743	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019738	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019707	C₂	1	2	yes	maximal, complemented
8	#26019700	C₂	1	2	yes	maximal, complemented
9	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018962	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152132	C₂²	1	1	yes	characteristic
13	#19151997	—	2	2	no	—
14	#19151955	—	2	2	no	—
15	#19151939	—	2	2	no	—
16	#19151922	—	2	2	no	—
17	#19151890	C₂²	1	2	yes	—
18	#19151885	C₂²	1	2	yes	—
19	#19151880	C₂²	1	2	yes	—
20	#19151875	C₂²	1	2	yes	—
21	#19151728	C₂²	1	1	yes	characteristic
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	#19151677	C₂²	1	1	yes	characteristic
24	#19151669	—	2	1	no	—
25	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
26	C₈×C₂²⋊D₈₉	—	2	1	no	—
27	#19151657	—	2	1	no	—
28	#19151655	C₂²	1	1	yes	characteristic
29	#19151651	C₂²	1	2	yes	—
30	#19151552	—	2	1	no	—
31	#19151351	C₂²	1	2	yes	—
32	#19151349	C₂²	1	1	yes	characteristic
33	#19151349	C₂²	1	1	yes	characteristic
34	#19151347	C₂²	1	2	yes	—
35	#19151347	C₂²	1	2	yes	—
36	#19151344	C₂²	1	2	yes	—
37	#19151344	C₂²	1	2	yes	—
38	#533×C₈₉	C₂²	1	2	yes	—
39	#500×C₈₉	C₂²	1	1	yes	characteristic
40	#494×C₈₉	C₂²	1	1	yes	characteristic
41	#487×C₈₉	C₂²	1	2	yes	—
42	#467×C₈₉	—	2	1	no	—
43	#467×C₈₉	—	2	1	no	—
44	C₄×Q₁₆×C₈₉	—	2	1	no	—
45	C₄×Q₁₆×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×#13083557	—	2	1	no	—
48	C₂×C₈×D₈₉	—	2	1	no	—
49	#15668006	—	2	1	no	—
50	#15668004	D₄	1	1	yes	characteristic
51	#15668000	D₄	1	1	yes	characteristic
52	#15667995	—	2	1	no	—
53	#15667991	D₄	1	1	yes	characteristic
54	#15667986	D₄	1	1	yes	characteristic
55	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
56	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
57	C₄×C₂²⋊D₈₉	—	2	1	no	—
58	#15667969	—	2	1	no	—
59	#15667945	—	2	2	no	—
60	#15667945	—	2	2	no	—
61	#15667943	—	2	2	no	—
62	#15667943	—	2	2	no	—
63	C₄⋊#10623494	C₂³	1	2	yes	—
64	C₄⋊#10623494	C₂³	1	2	yes	—
65	C₄⋊#10623494	C₂³	1	1	yes	characteristic
66	C₄⋊#10623494	C₂³	1	1	yes	characteristic
67	#15667941	—	2	1	no	—
68	#15667941	—	2	1	no	—
69	#15667935	—	2	2	no	—
70	#15667935	—	2	2	no	—
71	#15667932	—	2	2	no	—
72	#15667932	—	2	2	no	—
73	C₂²⋊#10623494	—	2	2	no	—
74	C₂²⋊#10623494	—	2	1	no	—
75	#15667924	—	2	1	no	—
76	#15667924	—	2	1	no	—
77	#15667923	—	2	1	no	—
78	#15667922	—	2	1	no	—
79	C₄×#10623494	—	2	2	no	—
80	C₈×Dic₈₉	—	2	1	no	—
81	C₂×Q₁₆×C₈₉	—	2	1	no	—
82	C₂×Q₁₆×C₈₉	—	2	1	no	—
83	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
84	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
85	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
87	#125×C₈₉	—	2	2	no	—
88	C₂²⋊Q₈×C₈₉	—	2	2	no	—
89	C₂²⋊Q₈×C₈₉	—	2	2	no	—
90	C₄×Q₈×C₈₉	—	2	2	no	—
91	C₄×Q₈×C₈₉	—	2	2	no	—
92	#117×C₈₉	—	2	2	no	—
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	#107×C₈₉	—	2	1	no	—
95	#107×C₈₉	—	2	1	no	—
96	#103×C₈₉	—	2	2	no	—
97	#103×C₈₉	—	2	2	no	—
98	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
99	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
100	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
103	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₈₉	—	2	1	no	—
105	C₂×C₄×D₈₉	—	2	1	no	—
106	#13083557	—	4	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	—	2	2	no	—
110	C₂×#10623494	—	2	2	no	—
111	C₈×D₈₉	—	4	2	no	—
112	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
114	#13083553	—	2	1	no	—
115	#13083553	—	2	1	no	—
116	#13083553	—	2	1	no	—
117	#13083553	—	2	1	no	—
118	C₂²⋊Dic₈₉	—	2	1	no	—
119	#13083551	—	2	1	no	—
120	#13083551	—	2	1	no	—
121	#13083551	—	2	1	no	—
122	#13083551	—	2	1	no	—
123	C₄×Dic₈₉	—	2	1	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	2	no	—
126	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
129	Q₁₆×C₈₉	—	4	2	no	—
130	Q₁₆×C₈₉	—	4	2	no	—
131	C₂×C₈×C₈₉	—	2	2	no	—
132	C₂×C₈×C₈₉	—	4	1	no	—
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₈₉	—	2	2	no	—
138	C₄⋊C₄×C₈₉	—	2	2	no	—
139	C₄⋊C₄×C₈₉	—	2	2	no	—
140	C₄⋊C₄×C₈₉	—	2	2	no	—
141	C₄⋊C₄×C₈₉	—	4	2	no	—
142	C₄⋊C₄×C₈₉	—	4	2	no	—
143	C₂²⋊C₄×C₈₉	—	4	2	no	—
144	C₄²×C₈₉	—	4	2	no	—
145	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₈₉	#41	1	1	yes	characteristic
148	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₂³×C₈₉	#121	1	1	yes	characteristic
150	C₂²×D₈₉	—	2	1	no	—
151	Q₈×C₈₉	—	4	2	no	—
152	Q₈×C₈₉	—	4	2	no	—
153	Q₈×C₈₉	—	8	2	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	2	1	no	—
156	C₂×C₄×C₈₉	—	2	2	no	—
157	C₂×C₄×C₈₉	—	4	1	no	—
158	C₂×C₄×C₈₉	—	4	2	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
161	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
162	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
166	C₂²⋊D₈₉	—	4	2	no	—
167	C₂×Dic₈₉	—	2	1	no	—
168	C₂×Dic₈₉	—	2	1	no	—
169	C₂×Dic₈₉	—	2	1	no	—
170	D₃₅₆	—	2	2	no	—
171	C₄×D₈₉	—	4	1	no	—
172	C₄×D₈₉	—	4	1	no	—
173	Dic₁₇₈	—	2	2	no	—
174	C₈×C₈₉	—	2	1	no	—
175	C₈×C₈₉	—	2	1	no	—
176	C₈×C₈₉	—	2	1	no	—
177	C₈×C₈₉	#135	1	2	yes	—
178	#10623494	—	4	1	no	—
179	#10623494	—	4	1	no	—
180	#10623494	—	4	2	no	—
181	#10623494	—	4	2	no	—
182	C₂²×C₈₉	—	2	1	no	—
183	C₂²×C₈₉	—	2	1	no	—
184	C₂²×C₈₉	—	2	1	no	—
185	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
186	C₂×D₈₉	—	4	1	no	—
187	C₂×D₈₉	—	4	1	no	—
188	C₂×D₈₉	—	4	1	no	—
189	C₄×C₈₉	—	2	1	no	—
190	C₄×C₈₉	—	2	1	no	—
191	C₄×C₈₉	—	2	1	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	8	2	no	—
194	C₄×C₈₉	—	8	2	no	—
195	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
196	C₄×C₈₉	#575	1	1	yes	characteristic
197	C₄×C₈₉	#578	1	2	yes	—
198	Dic₈₉	—	4	1	no	—
199	Dic₈₉	—	4	1	no	—
200	Dic₈₉	—	4	1	no	—
201	#19226	—	89	1	no	maximal, Sylow
202	C₂×C₈₉	—	4	1	no	—
203	C₂×C₈₉	#3035	1	1	yes	characteristic
204	C₂×C₈₉	#3059	1	1	yes	characteristic
205	C₂×C₈₉	#3067	1	1	yes	characteristic
206	D₈₉	—	8	1	no	—
207	#3233	—	89	2	no	—
208	#3170	D₈₉	1	1	yes	characteristic, complemented
209	#2966	—	89	1	no	—
210	#1696	—	89	1	no	—
211	#1695	—	89	1	no	—
212	#1690	—	89	1	no	—
213	#1689	—	89	1	no	—
214	#1684	—	89	2	no	—
215	#1683	—	89	2	no	—
216	#1564	—	89	1	no	—
217	#1554	—	89	2	no	—
218	C₈₉	#19226	1	1	yes	characteristic, complemented, Sylow
219	#533	C₂×D₈₉	1	2	yes	—
220	#529	—	89	1	no	—
221	#500	C₂×D₈₉	1	1	yes	characteristic
222	#499	—	178	2	no	—
223	#494	C₂×D₈₉	1	1	yes	characteristic
224	#488	—	178	2	no	—
225	#487	C₂×D₈₉	1	2	yes	—
226	#485	—	178	2	no	—
227	#482	—	178	2	no	—
228	#477	—	89	2	no	—
229	#476	—	89	2	no	—
230	#475	—	89	2	no	—
231	#474	—	89	2	no	—
232	#467	—	2	1	no	—
233	#467	—	2	1	no	—
234	#446	—	89	1	no	—
235	C₈×Q₈	—	89	1	no	—
236	C₄×Q₁₆	—	2	1	no	—
237	C₄×Q₁₆	—	2	1	no	—
238	#436	—	89	1	no	—
239	#435	—	178	1	no	—
240	C₈×D₄	—	89	1	no	—
241	C₈×D₄	—	178	1	no	—
242	#433	—	178	1	no	—
243	#432	—	89	1	no	—
244	#431	—	89	2	no	—
245	#405	—	178	1	no	—
246	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
247	#335	—	89	2	no	—
248	#334	—	89	1	no	—
249	#334	—	89	1	no	—
250	#333	—	89	2	no	—
251	#333	—	89	2	no	—
252	#332	—	89	2	no	—
253	#332	—	89	2	no	—
254	C₂×Q₁₆	—	2	1	no	—
255	C₂×Q₁₆	—	2	1	no	—
256	C₂×M₁₆	—	178	1	no	—
257	C₂²×C₈	—	178	1	no	—
258	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
260	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
261	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
262	#126	—	178	1	no	—
263	#125	—	2	2	no	—
264	#125	—	89	1	no	—
265	#124	—	89	1	no	—
266	#123	—	178	1	no	—
267	C₂²⋊Q₈	—	2	2	no	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₂²⋊Q₈	—	89	1	no	—
270	#121	—	89	1	no	—
271	C₄×Q₈	—	2	2	no	—
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	89	1	no	—
274	C₄×D₄	—	89	1	no	—
275	C₄×D₄	—	178	1	no	—
276	#117	—	2	2	no	—
277	#117	—	178	1	no	—
278	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
279	#107	—	2	1	no	—
280	#107	—	2	1	no	—
281	#107	—	178	2	no	—
282	#107	—	178	2	no	—
283	#106	—	178	2	no	—
284	#106	—	178	2	no	—
285	C₄⋊C₈	—	89	1	no	—
286	C₄⋊C₈	—	89	1	no	—
287	C₄⋊C₈	—	89	2	no	—
288	C₄⋊C₈	—	89	2	no	—
289	C₄⋊C₈	—	178	1	no	—
290	C₄⋊C₈	—	178	1	no	—
291	#103	—	2	2	no	—
292	#103	—	2	2	no	—
293	#103	—	178	2	no	—
294	#103	—	178	2	no	—
295	#102	—	178	2	no	—
296	#102	—	178	2	no	—
297	C₂²⋊C₈	—	178	2	no	—
298	C₂²⋊C₈	—	178	1	no	—
299	C₂²⋊C₈	—	178	1	no	—
300	C₂²⋊C₈	—	178	1	no	—
301	#97	—	178	1	no	—
302	#97	—	178	1	no	—
303	C₄×C₈	—	178	1	no	—
304	C₄×C₈	—	178	2	no	—
305	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
306	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
307	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
308	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	89	1	no	—
312	C₂×D₄	—	89	1	no	—
313	C₂×D₄	—	178	1	no	—
314	C₂²×C₄	—	178	1	no	—
315	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
316	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
317	C₂²×C₄	#13083573	1	1	yes	characteristic
318	Q₁₆	—	4	2	no	—
319	Q₁₆	—	4	2	no	—
320	M₁₆	—	356	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	356	2	no	—
324	C₂×C₈	—	178	2	no	—
325	C₂×C₈	—	178	2	no	—
326	C₂×C₈	—	178	1	no	—
327	C₂×C₈	—	178	1	no	—
328	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
330	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
331	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	89	1	no	—
339	C₄⋊C₄	—	89	1	no	—
340	C₄⋊C₄	—	178	1	no	—
341	C₄⋊C₄	—	178	1	no	—
342	C₄⋊C₄	—	178	1	no	—
343	C₄⋊C₄	—	178	1	no	—
344	C₂²⋊C₄	—	4	2	no	—
345	C₂²⋊C₄	—	178	1	no	—
346	C₂²⋊C₄	—	178	1	no	—
347	C₂²⋊C₄	—	178	1	no	—
348	C₂²⋊C₄	—	178	1	no	—
349	C₂²⋊C₄	—	178	1	no	—
350	C₄²	—	4	2	no	—
351	C₄²	—	178	1	no	—
352	C₄²	D₄×D₈₉	1	1	yes	characteristic
353	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
354	C₄²	#13083573	1	1	yes	characteristic
355	C₄²	C₂³×D₈₉	1	1	yes	characteristic
356	C₂³	—	178	1	no	—
357	C₂³	#15667983	1	1	yes	characteristic
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	8	2	no	—
361	Q₈	—	178	2	no	—
362	D₄	—	356	2	no	—
363	D₄	—	178	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	356	1	no	—
371	C₂×C₄	—	356	1	no	—
372	C₂×C₄	—	178	1	no	—
373	C₂×C₄	—	178	1	no	—
374	C₂×C₄	—	178	1	no	—
375	C₂×C₄	#15667983	1	1	yes	characteristic
376	C₂×C₄	#15667983	1	1	yes	characteristic
377	C₂×C₄	#15667983	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
379	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
380	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
381	C₈	—	2	1	no	—
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	356	1	no	—
385	C₈	—	356	2	no	—
386	C₈	—	356	2	no	—
387	C₈	—	356	1	no	—
388	C₈	#15668032	1	2	yes	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	356	1	no	—
393	C₂²	—	356	1	no	—
394	C₂²	—	356	1	no	—
395	C₂²	C₂×#15667983	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	8	2	no	—
401	C₄	—	8	2	no	—
402	C₄	—	356	1	no	—
403	C₄	—	356	1	no	—
404	C₄	—	356	1	no	—
405	C₄	C₂×#15668032	1	1	yes	characteristic
406	C₄	#19152396	1	1	yes	characteristic
407	C₄	#19152411	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	712	1	no	—
410	C₂	#26029701	1	1	yes	characteristic
411	C₂	#26029903	1	1	yes	characteristic
412	C₂	#26029968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3170	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#15667986	D₄
hypercenter	#3170	D₈₉
hypocenter	C₈₉	#19226
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #26019777	C₂	semidirect
4. #26019772	C₂	semidirect
5. #26019743	C₂	semidirect
6. #26019738	C₂	semidirect
7. #26019707	C₂	semidirect
8. #26019700	C₂	semidirect
9. #26019000	C₂	semidirect
10. #26018962	C₂	semidirect
11. C₈₉×#3170	C₂	semidirect
208. #3170	D₈₉	semidirect
218. C₈₉	#19226	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26031431	2
#26029227	2
#26019777	2
#26019772	2
#26019743	2
#26019738	2
#26019707	2
#26019700	2
#26019000	2
#26018962	2
C₈₉×#3170	2
#19226	89